{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PDE-FIND for identifying diffusion from a random walk\n",
    "\n",
    "Samuel Rudy, 2016\n",
    "\n",
    "In this notebook we demonstrate the method for finding the PDE associated with the distribution of a particle with stochastic trajectory.  The diffusion equation is derived from a random walk, meant to be discrete measurements of Brownian motion.  A time series of length $10^6$ is generated with $x_{n+1} \\sim \\mathcal{N}(x_n, dt)$.  From this we approximate the distribution function of the future potision of the trajectory and fit to a PDE.  In theory, we expect $f_t = 0.5f_{xx}$.  The algorithm achieves the correct PDE with parameter error $\\sim 10^{-3}$.  \n",
    "\n",
    "A second time series is used to try to identify the advection diffusion equation.  This time, $x_{n+1} \\sim \\mathcal{N}(x_n + c dt, dt)$.  We expect the distribution function to follow $f_t = 0.5f_{xx} - c f_x$.  Trying a few different sparsity promoting methods yields different results, with  greedy algorithm being the only one that works."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "pylab.rcParams['figure.figsize'] = (15,10)\n",
    "import numpy as np\n",
    "from PDE_FIND import *\n",
    "labelfontsize = 16; tickfontsize = 16; titlefontsize = 18"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data Collection\n",
    "\n",
    "First generate a dataset.  This is just a bunch of points where each is drawn from a normal distribution centered on it's predecesor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ -50.,    0.,   50.,  100.,  150.,  200.]),\n",
       " <a list of 6 Text yticklabel objects>)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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wuE1cU1SUXf3yjbU18fjj/Ss3162iyQwaZO7v2AEceywwaxYwd673Mpo29b1a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\nCyjjb3P58uT5Ro1KTMuHZWQ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qBJi/Hy+typ07B1ePdEQiMAUwGnrt0f4AFICnY8eDYudn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X1TT5e0UtJGobINE2VnxB0fr/p+SVorS9kRiX+YQxLHIxLHO4fqdJUfC31jqGxg4j++I0Nl7SVN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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f64fad2a890>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "length = 10**6\n",
    "dt = 0.01\n",
    "numpy.random.seed(0)\n",
    "pos = np.cumsum(np.sqrt(dt)*np.random.randn(length))\n",
    "    \n",
    "plot(dt*np.arange(length),pos, linewidth = 1.5)\n",
    "xlabel('Time', fontsize = labelfontsize)\n",
    "ylabel('Location', fontsize = labelfontsize)\n",
    "title('A single trace of Brownian motion', fontsize = titlefontsize)\n",
    "xticks(fontsize = tickfontsize); yticks(fontsize = tickfontsize)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finding the distribution function\n",
    "\n",
    "Now we take the single trajectory, split it up into many smaller ones, and create a histogram to approximate the distribution funciton.  The histograms clearly show a normal distribution spreading out as time goes on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f64f7128f90>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Dy5cv56KLLupzP5s2beK+++7js5/97BbHjz32WFatWlWXdh/96EeZOXMm73nP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\nkiRJrWd4lSRJkiS13v8H/7CbebwfM7cAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f650685f1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pylab.rcParams['figure.figsize'] = (15,10)\n",
    "\n",
    "P = {}\n",
    "M = 0\n",
    "\n",
    "m = 5\n",
    "n = 300\n",
    "\n",
    "for i in range(m):\n",
    "    P[i] = []\n",
    "    \n",
    "for i in range(len(pos)-m):\n",
    "    \n",
    "    # center\n",
    "    y = pos[i+1:i+m+1] - pos[i]\n",
    "    M = max([M, max(abs(y))])\n",
    "    \n",
    "    # add to distribution\n",
    "    for j in range(m):\n",
    "        P[j].append(y[j])\n",
    "    \n",
    "bins = np.linspace(-M,M,n+1)\n",
    "x = linspace(M*(1/n-1),M*(1-1/n),n)\n",
    "dx = x[2]-x[1]\n",
    "T = linspace(0,dt*(m-1),m)\n",
    "U = np.zeros((n,m))\n",
    "for i in range(m):\n",
    "    U[:,i] = hist(P[i],bins,label=r'$t = $' + str(i*dt+dt))[0]/float(dx*(len(pos)-m))\n",
    "    \n",
    "xlabel('Location', fontsize = labelfontsize)\n",
    "ylabel(r'$f(x,t)$', fontsize = labelfontsize)\n",
    "title(r'Histograms for $f(x,t)$', fontsize = titlefontsize)\n",
    "xticks(fontsize = tickfontsize); yticks(fontsize = tickfontsize)\n",
    "legend(loc = 'upper right', fontsize = 14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "source": [
    "## Using PDE-FIND to identify the diffusion equation\n",
    "\n",
    "Finally we derive a PDE for the distribution using PDE-FIND.  For simple Brownian motion, it works quite well.  Below, we show that an added diffusion term makes it more difficult to correctly idenitify the dynamics."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Candidate functions for PDE\n",
      "1\n",
      "u\n",
      "u^2\n",
      "u^3\n",
      "u^4\n",
      "u^5\n",
      "u_{x}\n",
      "uu_{x}\n",
      "u^2u_{x}\n",
      "u^3u_{x}\n",
      "u^4u_{x}\n",
      "u^5u_{x}\n",
      "u_{xx}\n",
      "uu_{xx}\n",
      "u^2u_{xx}\n",
      "u^3u_{xx}\n",
      "u^4u_{xx}\n",
      "u^5u_{xx}\n",
      "u_{xxx}\n",
      "uu_{xxx}\n",
      "u^2u_{xxx}\n",
      "u^3u_{xxx}\n",
      "u^4u_{xxx}\n",
      "u^5u_{xxx}\n",
      "u_{xxxx}\n",
      "uu_{xxxx}\n",
      "u^2u_{xxxx}\n",
      "u^3u_{xxxx}\n",
      "u^4u_{xxxx}\n",
      "u^5u_{xxxx}\n",
      "\n",
      "PDE derived from data:\n",
      "u_t = (0.498425 +0.000000i)u_{xx}\n",
      "   \n"
     ]
    }
   ],
   "source": [
    "Ut,R,rhs_des = build_linear_system(U, dt, dx, D=4, P=5, time_diff = 'FD', deg_x = 4)\n",
    "w = TrainSTRidge(R, Ut, 10**-2, 10, normalize = 2)\n",
    "\n",
    "print \"Candidate functions for PDE\"\n",
    "for func in ['1'] + rhs_des[1:]: print func\n",
    "\n",
    "print \"\\nPDE derived from data:\"\n",
    "print_pde(w, rhs_des)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Advection diffusion from a biased random walk\n",
    "\n",
    "Now we'll try adding an advection term.  Looking at the histograms, we see that they are similar but the means appear to be moving to the right due to the advection term."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f64f7791f90>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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5z21x/Oijj2b16tV1afexj32MOXPm8J73vKdPOa5cuZKVK1f2qe3WlLZ4jYiJ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\neJUkSZIkNd7/B4TuvY4pWggRAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f64fae2b250>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "length = 10**6\n",
    "dt = 0.01\n",
    "c = 2\n",
    "numpy.random.seed(0)\n",
    "pos = np.cumsum(np.sqrt(dt)*np.random.randn(length)) + c*dt*np.arange(length)\n",
    "\n",
    "P = {}\n",
    "M = 0\n",
    "\n",
    "m = 5\n",
    "n = 300\n",
    "\n",
    "for i in range(m):\n",
    "    P[i] = []\n",
    "    \n",
    "for i in range(len(pos)-m):\n",
    "    \n",
    "    # center\n",
    "    y = pos[i+1:i+m+1] - pos[i]\n",
    "    M = max([M, max(abs(y))])\n",
    "    \n",
    "    # add to distribution\n",
    "    for j in range(m):\n",
    "        P[j].append(y[j])\n",
    "    \n",
    "bins = np.linspace(-M,M,n+1)\n",
    "x = linspace(M*(1/n-1),M*(1-1/n),n)\n",
    "dx = x[2]-x[1]\n",
    "T = linspace(0,dt*(m-1),m)\n",
    "U = np.zeros((n,m))\n",
    "for i in range(m):\n",
    "    U[:,i] = hist(P[i],bins,label=r'$t = $' + str(i*dt+dt))[0]/float(dx*(len(pos)-m))\n",
    "    \n",
    "xlabel('Location', fontsize = labelfontsize)\n",
    "ylabel(r'$f(x,t)$', fontsize = labelfontsize)\n",
    "title(r'Histograms for $f(x,t)$', fontsize = titlefontsize)\n",
    "xticks(fontsize = tickfontsize); yticks(fontsize = tickfontsize)\n",
    "legend(loc = 'upper right', fontsize = 14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now try to identify the dynamics.  This is an example of how different sparsity promoting regression methods can behave differently.  Here the STRidge algorithm works only when the data is not normalized.  The greedy algorithm seems to be fairly robust to small changes, and Lasso works with some tuning (maybe only because we know the right answer).  Normally STRidge with $L^2$ normalization outperforms all the rest."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "PDE derived with STRidge and L^2 normalization (Correct PDE!)\n",
      "u_t = (-1.998342 +0.000000i)u_{x}\n",
      "    + (0.513308 +0.000000i)u_{xx}\n",
      "    + (0.000086 +0.000000i)u_{xxxx}\n",
      "   \n",
      "PDE derived with STRidge without normalization (Correct PDE!)\n",
      "u_t = (6.042714 +0.000000i)u\n",
      "    + (-4.986524 +0.000000i)u^2\n",
      "    + (0.756198 +0.000000i)u^3\n",
      "    + (-1.997343 +0.000000i)u_{x}\n",
      "    + (0.475029 +0.000000i)u_{xx}\n",
      "   \n",
      "PDE derived with greedy algorithm (Correct PDE!)\n",
      "u_t = (-2.000641 +0.000000i)u_{x}\n",
      "    + (0.503582 +0.000000i)u_{xx}\n",
      "   \n",
      "PDE derived with Lasso (Incorrect PDE)\n",
      "u_t = (0.531298 +0.000000i)u_{xx}\n",
      "    + (0.000270 +0.000000i)u_{xxxx}\n",
      "   \n"
     ]
    }
   ],
   "source": [
    "# Ut,R,rhs_des = build_linear_system(U, dt, dx, D=4, P=5, time_diff = 'FD', deg_x = 4)\n",
    "\n",
    "# print \"Candidate functions for PDE\"\n",
    "# for func in ['1'] + rhs_des[1:]: print func\n",
    "\n",
    "print \"\\nPDE derived with STRidge and L^2 normalization (Correct PDE!)\"\n",
    "w = TrainSTRidge(R, Ut, 1, 4, normalize = 2)\n",
    "print_pde(w, rhs_des)\n",
    "\n",
    "print \"PDE derived with STRidge without normalization (Correct PDE!)\"\n",
    "w = TrainSTRidge(R, Ut, 1, 0.01, normalize = 0)\n",
    "print_pde(w, rhs_des)\n",
    "\n",
    "print \"PDE derived with greedy algorithm (Correct PDE!)\"\n",
    "w = FoBaGreedy(R, Ut,10)\n",
    "print_pde(w, rhs_des)\n",
    "\n",
    "print \"PDE derived with Lasso (Incorrect PDE)\"\n",
    "w = Lasso(R, Ut, 350)\n",
    "print_pde(w, rhs_des)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
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